Drive Systems and Kinematics

Robotics Fundamentals 3 chose a motor, a gearbox and a wheel, and showed how much force and torque it takes to move a robot. This chapter is about how those wheels are arranged and what the arrangement lets the robot do. Two wheels driven at different speeds steer a robot; a car steers by turning its front wheels; a wheel with rollers round its rim lets a robot slide sideways. Each layout has a short piece of geometry attached, called its kinematics: the relationship between wheel speeds and the motion of the robot, with no forces in it. Kinematics works in both directions. Forward kinematics turns wheel speeds into robot motion, and when you add up that motion over time it becomes odometry, the robot's own running estimate of where it is. Inverse kinematics turns a wanted motion into the wheel speeds that produce it. We then look at why odometry drifts, compare tracks with wheels, and take a first look at legs.

Dimensions, slip factors, encoder counts and pressures in this chapter are typical, approximate or illustrative values from memory, chosen to make the arithmetic clear. Real robots vary, so measure yours (the chapter ends with how). The formulas themselves are standard geometry and are exact for the idealised models described.

1. Holonomic and non-holonomic motion

A robot on a flat floor has a pose of three numbers: its position (x, y) and its heading θ. Robotics Fundamentals 1 counted degrees of freedom, and a ground robot has three. What matters for a drive system is how many of those three the robot can change independently, at any moment.

• A holonomic drive (in the robotics sense) can move in any direction, and rotate, independently and instantly. It controls all three: forward speed, sideways speed and turning rate. Omni-wheel and mecanum robots are holonomic.
• A non-holonomic drive cannot move sideways. A differential-drive robot and a car each control only two velocities (forward speed and turning rate, or forward speed and steering angle). The constraint is that the wheels will not slide sideways, which in symbols says the velocity along the axle direction is zero.

A non-holonomic robot can still reach any pose, but it cannot go there directly. A car cannot slide into a parking space beside it: it must go forward and back with the wheels turned (parallel parking), which is the classic non-holonomic manoeuvre. A holonomic robot would simply slide across. The price of that freedom is extra complexity, lower grip and more motors.

2. The drive types

3. Differential drive: forward kinematics

Let the two wheels be a distance L apart (the track width, measured between the contact points) and have ground speeds v_R and v_L (metres per second at the rim, positive when driving the robot forward). The robot body moves at the average of the two, and it turns at the difference divided by the wheel spacing.

The table below fixes v_R at 0.50 m/s and slows the left wheel, with L = 0.15 m. Notice how quickly the radius shrinks once the wheel speeds differ a lot, and that a stopped wheel gives a pivot, not a spin.

4. Differential drive: inverse kinematics

Planners and controllers think in v and ω ("go at 0.4 m/s and turn at 2 rad/s"), but the motors need wheel speeds. Solving the two forward equations for the wheel speeds gives the inverse.

5. Encoders, distance and odometry

How does the robot know the wheel speeds? A wheel encoder produces a number of pulses (or ticks) for each turn of its shaft, and counting them measures how far the wheel has turned. Robotics Fundamentals 6 treats encoders properly (how they work, direction sensing, resolution); here we only need the arithmetic. An encoder is usually on the motor shaft, before the gearbox, so each wheel revolution corresponds to the gear ratio times as many ticks.

Resolution is therefore rarely the limit with a geared encoder: a tick is a third of a millimetre. The errors that matter come from the model (the wheel diameter and track width you assumed) and from the floor (slip), as section 6 shows.

The pose update

Odometry keeps a running pose (x, y, θ). At each update (typically 20 to 100 times a second) it reads how many ticks each wheel has counted since the last update, converts them to distances Δd_L and Δd_R, and moves the pose. Over one short step the wheel speeds are constant, so the robot moves along a circular arc (section 3), and the exact update follows from the arc geometry.

Here is the update as short, hardware-neutral Python-style pseudocode (real board code, with its timers and interrupts for the encoders, is left to the course Programming for Robotics):

6. Odometry drift: where the errors come from

Odometry assumes that the wheel diameter, the track width and the wheel-on-floor contact are exactly what the software believes. Each can be a little wrong, and each gives a different kind of error. The errors that come from the robot's own dimensions are systematic: the same every time, so they can be measured and removed. The errors from slip and bumps are random: different every time, so they can only be reduced (by feedback from sensors), not calibrated away.

Worked example 1: one tyre 1 % too big

A robot with L = 0.15 m is told to drive 3.0 m straight. The software believes both wheels are the same size, so it runs both until each encoder has counted a distance of 3.0 m. The right tyre is in fact 1 % bigger than the left one, so its wheel rolls 1 % further for the same number of ticks.

Worked example 2: track width 3 % out

The software uses L = 0.150 m but the true effective track width is 0.155 m (the tyre contact patches are further apart than the hubs). The robot is told to rotate on the spot through 360 degrees, which, by the software's belief, needs each wheel to roll once round a circle of radius L / 2, in opposite directions.

Calibration

Because these errors are systematic, two measured tests remove most of them. Do each several times on the real floor and average.

1. Straight line, for wheel diameters. Drive a long, measured straight line and measure how far each wheel's distance was wrong and which way the robot curved. Scale each wheel's effective diameter by (true distance / odometry distance); the curve vanishes once the two wheels agree.
2. On-the-spot turn, for track width. Command a rotation of several full turns and measure the real angle (a marked floor works). Then L(effective) = L(assumed) × (angle by odometry) / (true angle). In worked example 2: 0.150 × 360 / 348.4 = 0.155 m.

A well-known refinement, the UMBmark test, drives a square clockwise and then anticlockwise and uses the two sets of end-point errors to separate the diameter error from the track error (the details are from memory, so look the method up before relying on it). Even after calibration the random errors remain, so a long-running robot needs another source of position information.

7. Car-style (Ackermann) steering

When a car turns, all four wheels must travel round circles about the same point, or some wheel would have to slide sideways. The rear wheels are fixed, so the centre of the circle lies on the line through the rear axle, and each front wheel must point at right angles to the line from that centre to it. That means the two front wheels turn by different angles: the inner wheel more than the outer. A linkage that does this is called Ackermann steering, and many hobby robots use a simple version of it, or a single steering servo with the two front wheels turned equally (which is only approximately right and scrubs the tyres a little).

8. Omni and mecanum wheels: holonomic drives

Omni wheels

An omni wheel has small rollers around its rim, with their axes at right angles to the wheel's axle. The wheel drives the robot along the direction it points, but the rollers let it slide freely across that direction. With three omni wheels at 120 degrees to each other (a "kiwi" drive) or four at 90 degrees, the sum of the wheels' pushes can point anywhere. The standard inverse kinematics for a robot with wheels at angles φ_i (the angle of each wheel's position, measured from the robot's x axis; each wheel drives tangentially, counter-clockwise) and distance R_b from the centre is:

Mecanum wheels

A mecanum wheel has rollers set at 45 degrees to its axle, so the wheel's push on the floor points diagonally. Four of them, fitted so the roller pattern is a mirror image front to back and left to right, give the robot a push in any direction by choosing the four speeds: if the diagonal components cancel, the robot goes forward; if the forward components cancel, it goes sideways.

Holonomic drives are attractive but have costs. Mecanum and omni wheels are less efficient and grip less than ordinary tyres (the rollers slip a little in the very act of moving sideways), so their odometry is more error-prone and they push less hard (Robotics Fundamentals 3 gave traction as F_max = μ N); mecanum wheels work well only on flat, smooth floors; and each wheel needs its own motor and speed control. They suit a robot that must move sideways in a tight space, such as a competition robot lining up on a target. A differential drive remains the better choice for most others.

9. Skid-steer, tracks and wheels

A skid-steer robot (four or six wheels, or two tracks, with the left side driven separately from the right side) has no caster and no steering. To turn, the two sides run at different speeds, and since the wheels or tracks cannot turn to follow the curve, they slide sideways over the ground. The simple differential formulas from section 3 then misreport the turn, because the effective turning geometry is wider than the geometry of the wheels.

The choice follows the terrain: wheels indoors and on hard flat ground, tracks (or large, grippy wheels, or a skid-steer with a lot of grip) outdoors on soft or loose ground. Four-wheel skid-steer robots are a common middle way: easy to build and tough, with the same odometry caveat.

10. Legs, briefly

Legs deal with ground that wheels cannot: stairs, rubble, gaps, a stepping stone. A leg only needs isolated footholds, where a wheel needs a continuous surface. The cost is complexity. A walking robot typically needs about three actuators per leg (a hip swing, a hip lift and a knee), so the actuator count and the control problem grow quickly.

• Gait. A gait is the repeating pattern in which legs lift and land. A hexapod's usual tripod gait moves three legs at once (the front and back legs on one side and the middle leg on the other) while the other three support the body, then swaps. The body is always supported by a triangle of feet.
• Static stability. A robot is statically stable when its centre of gravity (Robotics Fundamentals 2) lies over the polygon formed by the feet on the ground. Hexapods and slow quadrupeds can stay inside this polygon all the time; faster walking and running rely on dynamic balance, in which the robot is only stable because it keeps moving.
• Kinematics. A leg is a small arm: its foot position follows from its joint angles by the same forward and inverse kinematics as a robot arm, and Robotics Fundamentals 9 works through them for a two-link arm, which is the basis of a leg.
• Cost. Legs are slower and use more energy per metre than wheels on flat ground, because they lift and lower the body at every step. They are worth it only when the terrain demands them.

11. Challenges

Next, Robotics Fundamentals 5 turns to sensors, the robot's way of seeing the world: distance sensors (ultrasonic, infrared, time-of-flight and lidar), line, bump and light sensors, a first look at cameras, and the ideas of accuracy, noise and range that decide which sensor to choose. Odometry told the robot where it thinks it is; sensors begin to tell it what is actually around it.